Maybe this was just a one-time lucky event, but I replaced the batteries in my Magic Mouse this afternoon.
I switched it off, replaced the batteries and switched it back on. I clicked it once, and it was connected instantly.
That's the first time that's ever happened to me. Usually it takes some time, and several mouse clicks to get the mouse connected. And sometimes in the past, nothing I did would get it connected until I used my Magic Trackpad to choose the Mouse in the BlueTooth menu and then to choose Connect.

Maybe 10.10.2 fixed something in BlueTooth.

"Design is not just what it looks like and feels like. Design is how it works."
Steve Jobs